Integrand size = 25, antiderivative size = 290 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=-\frac {304 b^2 d x}{3675 c^4}-\frac {152 b^2 d x^3}{11025 c^2}-\frac {38 b^2 d x^5}{6125}+\frac {2}{343} b^2 c^2 d x^7+\frac {32 b d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{525 c^5}+\frac {16 b d x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{525 c^3}+\frac {4 b d x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{175 c}+\frac {2 b d \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{21 c^5}-\frac {4 b d \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{35 c^5}+\frac {2 b d \left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{49 c^5}+\frac {2}{35} d x^5 (a+b \arcsin (c x))^2+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2 \]
-304/3675*b^2*d*x/c^4-152/11025*b^2*d*x^3/c^2-38/6125*b^2*d*x^5+2/343*b^2* c^2*d*x^7+2/21*b*d*(-c^2*x^2+1)^(3/2)*(a+b*arcsin(c*x))/c^5-4/35*b*d*(-c^2 *x^2+1)^(5/2)*(a+b*arcsin(c*x))/c^5+2/49*b*d*(-c^2*x^2+1)^(7/2)*(a+b*arcsi n(c*x))/c^5+2/35*d*x^5*(a+b*arcsin(c*x))^2+1/7*d*x^5*(-c^2*x^2+1)*(a+b*arc sin(c*x))^2+32/525*b*d*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^5+16/525*b*d *x^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/c^3+4/175*b*d*x^4*(a+b*arcsin(c* x))*(-c^2*x^2+1)^(1/2)/c
Time = 0.18 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.70 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=-\frac {d \left (11025 a^2 c^5 x^5 \left (-7+5 c^2 x^2\right )+210 a b \sqrt {1-c^2 x^2} \left (-152-76 c^2 x^2-57 c^4 x^4+75 c^6 x^6\right )+b^2 \left (31920 c x+5320 c^3 x^3+2394 c^5 x^5-2250 c^7 x^7\right )+210 b \left (105 a c^5 x^5 \left (-7+5 c^2 x^2\right )+b \sqrt {1-c^2 x^2} \left (-152-76 c^2 x^2-57 c^4 x^4+75 c^6 x^6\right )\right ) \arcsin (c x)+11025 b^2 c^5 x^5 \left (-7+5 c^2 x^2\right ) \arcsin (c x)^2\right )}{385875 c^5} \]
-1/385875*(d*(11025*a^2*c^5*x^5*(-7 + 5*c^2*x^2) + 210*a*b*Sqrt[1 - c^2*x^ 2]*(-152 - 76*c^2*x^2 - 57*c^4*x^4 + 75*c^6*x^6) + b^2*(31920*c*x + 5320*c ^3*x^3 + 2394*c^5*x^5 - 2250*c^7*x^7) + 210*b*(105*a*c^5*x^5*(-7 + 5*c^2*x ^2) + b*Sqrt[1 - c^2*x^2]*(-152 - 76*c^2*x^2 - 57*c^4*x^4 + 75*c^6*x^6))*A rcSin[c*x] + 11025*b^2*c^5*x^5*(-7 + 5*c^2*x^2)*ArcSin[c*x]^2))/c^5
Time = 1.26 (sec) , antiderivative size = 331, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5202, 5138, 5194, 27, 2009, 5210, 15, 5210, 15, 5182, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^4 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx\) |
| \(\Big \downarrow \) 5202 |
| \(\displaystyle -\frac {2}{7} b c d \int x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {2}{7} d \int x^4 (a+b \arcsin (c x))^2dx+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{7} b c d \int x^5 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))dx+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 5194 |
| \(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{7} b c d \left (-b c \int -\frac {-15 c^6 x^6+3 c^4 x^4+4 c^2 x^2+8}{105 c^6}dx-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}\right )+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )-\frac {2}{7} b c d \left (\frac {b \int \left (-15 c^6 x^6+3 c^4 x^4+4 c^2 x^2+8\right )dx}{105 c^5}-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}\right )+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \int \frac {x^5 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx\right )+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c d \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {4 \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{5 c^2}+\frac {b \int x^4dx}{5 c}-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}\right )\right )+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c d \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {4 \int \frac {x^3 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{5 c^2}-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c d \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}+\frac {b \int x^2dx}{3 c}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}\right )}{5 c^2}-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c d \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {4 \left (\frac {2 \int \frac {x (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )}{5 c^2}-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c d \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\) |
| \(\Big \downarrow \) 5182 |
| \(\displaystyle \frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (\frac {4 \left (\frac {2 \left (\frac {b \int 1dx}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{3 c^2}-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {b x^3}{9 c}\right )}{5 c^2}-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {b x^5}{25 c}\right )\right )+\frac {1}{7} d x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2-\frac {2}{7} b c d \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{7} d x^5 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {2}{7} d \left (\frac {1}{5} x^5 (a+b \arcsin (c x))^2-\frac {2}{5} b c \left (-\frac {x^4 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{5 c^2}+\frac {4 \left (-\frac {x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 c^2}+\frac {2 \left (\frac {b x}{c}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c^2}\right )}{3 c^2}+\frac {b x^3}{9 c}\right )}{5 c^2}+\frac {b x^5}{25 c}\right )\right )-\frac {2}{7} b c d \left (-\frac {\left (1-c^2 x^2\right )^{7/2} (a+b \arcsin (c x))}{7 c^6}+\frac {2 \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{5 c^6}-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{3 c^6}+\frac {b \left (-\frac {15}{7} c^6 x^7+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\right )\) |
(d*x^5*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/7 - (2*b*c*d*((b*(8*x + (4*c^2 *x^3)/3 + (3*c^4*x^5)/5 - (15*c^6*x^7)/7))/(105*c^5) - ((1 - c^2*x^2)^(3/2 )*(a + b*ArcSin[c*x]))/(3*c^6) + (2*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin[c*x] ))/(5*c^6) - ((1 - c^2*x^2)^(7/2)*(a + b*ArcSin[c*x]))/(7*c^6)))/7 + (2*d* ((x^5*(a + b*ArcSin[c*x])^2)/5 - (2*b*c*((b*x^5)/(25*c) - (x^4*Sqrt[1 - c^ 2*x^2]*(a + b*ArcSin[c*x]))/(5*c^2) + (4*((b*x^3)/(9*c) - (x^2*Sqrt[1 - c^ 2*x^2]*(a + b*ArcSin[c*x]))/(3*c^2) + (2*((b*x)/c - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c^2))/(3*c^2)))/(5*c^2)))/5))/7
3.2.56.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ .), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] I nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_) , x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSin [c*x]) u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int[Sim plifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcS in[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f*x) ^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 - c^2 *x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(e*(m + 2*p + 1))), x] + (Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] + S imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Int[(f* x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m , 1] && NeQ[m + 2*p + 1, 0]
Time = 0.24 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.95
| method | result | size |
| parts | \(-d \,a^{2} \left (\frac {1}{7} c^{2} x^{7}-\frac {1}{5} x^{5}\right )-\frac {d \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {2 \arcsin \left (c x \right ) \left (3 c^{4} x^{4}+4 c^{2} x^{2}+8\right ) \sqrt {-c^{2} x^{2}+1}}{75}+\frac {38 c^{5} x^{5}}{6125}+\frac {152 c^{3} x^{3}}{11025}+\frac {304 c x}{3675}+\frac {\arcsin \left (c x \right )^{2} c^{7} x^{7}}{7}+\frac {2 \arcsin \left (c x \right ) \left (5 c^{6} x^{6}+6 c^{4} x^{4}+8 c^{2} x^{2}+16\right ) \sqrt {-c^{2} x^{2}+1}}{245}-\frac {2 c^{7} x^{7}}{343}\right )}{c^{5}}-\frac {2 d a b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {19 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {76 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {-c^{2} x^{2}+1}}{3675}\right )}{c^{5}}\) | \(275\) |
| derivativedivides | \(\frac {-d \,a^{2} \left (\frac {1}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {2 \arcsin \left (c x \right ) \left (3 c^{4} x^{4}+4 c^{2} x^{2}+8\right ) \sqrt {-c^{2} x^{2}+1}}{75}+\frac {38 c^{5} x^{5}}{6125}+\frac {152 c^{3} x^{3}}{11025}+\frac {304 c x}{3675}+\frac {\arcsin \left (c x \right )^{2} c^{7} x^{7}}{7}+\frac {2 \arcsin \left (c x \right ) \left (5 c^{6} x^{6}+6 c^{4} x^{4}+8 c^{2} x^{2}+16\right ) \sqrt {-c^{2} x^{2}+1}}{245}-\frac {2 c^{7} x^{7}}{343}\right )-2 d a b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {19 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {76 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {-c^{2} x^{2}+1}}{3675}\right )}{c^{5}}\) | \(276\) |
| default | \(\frac {-d \,a^{2} \left (\frac {1}{7} c^{7} x^{7}-\frac {1}{5} c^{5} x^{5}\right )-d \,b^{2} \left (-\frac {\arcsin \left (c x \right )^{2} c^{5} x^{5}}{5}-\frac {2 \arcsin \left (c x \right ) \left (3 c^{4} x^{4}+4 c^{2} x^{2}+8\right ) \sqrt {-c^{2} x^{2}+1}}{75}+\frac {38 c^{5} x^{5}}{6125}+\frac {152 c^{3} x^{3}}{11025}+\frac {304 c x}{3675}+\frac {\arcsin \left (c x \right )^{2} c^{7} x^{7}}{7}+\frac {2 \arcsin \left (c x \right ) \left (5 c^{6} x^{6}+6 c^{4} x^{4}+8 c^{2} x^{2}+16\right ) \sqrt {-c^{2} x^{2}+1}}{245}-\frac {2 c^{7} x^{7}}{343}\right )-2 d a b \left (\frac {\arcsin \left (c x \right ) c^{7} x^{7}}{7}-\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}+\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{49}-\frac {19 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{1225}-\frac {76 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3675}-\frac {152 \sqrt {-c^{2} x^{2}+1}}{3675}\right )}{c^{5}}\) | \(276\) |
-d*a^2*(1/7*c^2*x^7-1/5*x^5)-d*b^2/c^5*(-1/5*arcsin(c*x)^2*c^5*x^5-2/75*ar csin(c*x)*(3*c^4*x^4+4*c^2*x^2+8)*(-c^2*x^2+1)^(1/2)+38/6125*c^5*x^5+152/1 1025*c^3*x^3+304/3675*c*x+1/7*arcsin(c*x)^2*c^7*x^7+2/245*arcsin(c*x)*(5*c ^6*x^6+6*c^4*x^4+8*c^2*x^2+16)*(-c^2*x^2+1)^(1/2)-2/343*c^7*x^7)-2*d*a*b/c ^5*(1/7*arcsin(c*x)*c^7*x^7-1/5*arcsin(c*x)*c^5*x^5+1/49*c^6*x^6*(-c^2*x^2 +1)^(1/2)-19/1225*c^4*x^4*(-c^2*x^2+1)^(1/2)-76/3675*c^2*x^2*(-c^2*x^2+1)^ (1/2)-152/3675*(-c^2*x^2+1)^(1/2))
Time = 0.28 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.79 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=-\frac {1125 \, {\left (49 \, a^{2} - 2 \, b^{2}\right )} c^{7} d x^{7} - 63 \, {\left (1225 \, a^{2} - 38 \, b^{2}\right )} c^{5} d x^{5} + 5320 \, b^{2} c^{3} d x^{3} + 31920 \, b^{2} c d x + 11025 \, {\left (5 \, b^{2} c^{7} d x^{7} - 7 \, b^{2} c^{5} d x^{5}\right )} \arcsin \left (c x\right )^{2} + 22050 \, {\left (5 \, a b c^{7} d x^{7} - 7 \, a b c^{5} d x^{5}\right )} \arcsin \left (c x\right ) + 210 \, {\left (75 \, a b c^{6} d x^{6} - 57 \, a b c^{4} d x^{4} - 76 \, a b c^{2} d x^{2} - 152 \, a b d + {\left (75 \, b^{2} c^{6} d x^{6} - 57 \, b^{2} c^{4} d x^{4} - 76 \, b^{2} c^{2} d x^{2} - 152 \, b^{2} d\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{385875 \, c^{5}} \]
-1/385875*(1125*(49*a^2 - 2*b^2)*c^7*d*x^7 - 63*(1225*a^2 - 38*b^2)*c^5*d* x^5 + 5320*b^2*c^3*d*x^3 + 31920*b^2*c*d*x + 11025*(5*b^2*c^7*d*x^7 - 7*b^ 2*c^5*d*x^5)*arcsin(c*x)^2 + 22050*(5*a*b*c^7*d*x^7 - 7*a*b*c^5*d*x^5)*arc sin(c*x) + 210*(75*a*b*c^6*d*x^6 - 57*a*b*c^4*d*x^4 - 76*a*b*c^2*d*x^2 - 1 52*a*b*d + (75*b^2*c^6*d*x^6 - 57*b^2*c^4*d*x^4 - 76*b^2*c^2*d*x^2 - 152*b ^2*d)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^5
Time = 0.88 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.34 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=\begin {cases} - \frac {a^{2} c^{2} d x^{7}}{7} + \frac {a^{2} d x^{5}}{5} - \frac {2 a b c^{2} d x^{7} \operatorname {asin}{\left (c x \right )}}{7} - \frac {2 a b c d x^{6} \sqrt {- c^{2} x^{2} + 1}}{49} + \frac {2 a b d x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {38 a b d x^{4} \sqrt {- c^{2} x^{2} + 1}}{1225 c} + \frac {152 a b d x^{2} \sqrt {- c^{2} x^{2} + 1}}{3675 c^{3}} + \frac {304 a b d \sqrt {- c^{2} x^{2} + 1}}{3675 c^{5}} - \frac {b^{2} c^{2} d x^{7} \operatorname {asin}^{2}{\left (c x \right )}}{7} + \frac {2 b^{2} c^{2} d x^{7}}{343} - \frac {2 b^{2} c d x^{6} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{49} + \frac {b^{2} d x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} - \frac {38 b^{2} d x^{5}}{6125} + \frac {38 b^{2} d x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{1225 c} - \frac {152 b^{2} d x^{3}}{11025 c^{2}} + \frac {152 b^{2} d x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3675 c^{3}} - \frac {304 b^{2} d x}{3675 c^{4}} + \frac {304 b^{2} d \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{3675 c^{5}} & \text {for}\: c \neq 0 \\\frac {a^{2} d x^{5}}{5} & \text {otherwise} \end {cases} \]
Piecewise((-a**2*c**2*d*x**7/7 + a**2*d*x**5/5 - 2*a*b*c**2*d*x**7*asin(c* x)/7 - 2*a*b*c*d*x**6*sqrt(-c**2*x**2 + 1)/49 + 2*a*b*d*x**5*asin(c*x)/5 + 38*a*b*d*x**4*sqrt(-c**2*x**2 + 1)/(1225*c) + 152*a*b*d*x**2*sqrt(-c**2*x **2 + 1)/(3675*c**3) + 304*a*b*d*sqrt(-c**2*x**2 + 1)/(3675*c**5) - b**2*c **2*d*x**7*asin(c*x)**2/7 + 2*b**2*c**2*d*x**7/343 - 2*b**2*c*d*x**6*sqrt( -c**2*x**2 + 1)*asin(c*x)/49 + b**2*d*x**5*asin(c*x)**2/5 - 38*b**2*d*x**5 /6125 + 38*b**2*d*x**4*sqrt(-c**2*x**2 + 1)*asin(c*x)/(1225*c) - 152*b**2* d*x**3/(11025*c**2) + 152*b**2*d*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(3675 *c**3) - 304*b**2*d*x/(3675*c**4) + 304*b**2*d*sqrt(-c**2*x**2 + 1)*asin(c *x)/(3675*c**5), Ne(c, 0)), (a**2*d*x**5/5, True))
Time = 0.29 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.56 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=-\frac {1}{7} \, b^{2} c^{2} d x^{7} \arcsin \left (c x\right )^{2} - \frac {1}{7} \, a^{2} c^{2} d x^{7} + \frac {1}{5} \, b^{2} d x^{5} \arcsin \left (c x\right )^{2} + \frac {1}{5} \, a^{2} d x^{5} - \frac {2}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} a b c^{2} d - \frac {2}{25725} \, {\left (105 \, {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c \arcsin \left (c x\right ) - \frac {75 \, c^{6} x^{7} + 126 \, c^{4} x^{5} + 280 \, c^{2} x^{3} + 1680 \, x}{c^{6}}\right )} b^{2} c^{2} d + \frac {2}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b d + \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arcsin \left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} d \]
-1/7*b^2*c^2*d*x^7*arcsin(c*x)^2 - 1/7*a^2*c^2*d*x^7 + 1/5*b^2*d*x^5*arcsi n(c*x)^2 + 1/5*a^2*d*x^5 - 2/245*(35*x^7*arcsin(c*x) + (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*a*b*c^2*d - 2/25725*(105*(5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c*arcsin(c*x) - (75*c^6*x^7 + 126*c^4*x^5 + 2 80*c^2*x^3 + 1680*x)/c^6)*b^2*c^2*d + 2/75*(15*x^5*arcsin(c*x) + (3*sqrt(- c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1) /c^6)*c)*a*b*d + 2/1125*(15*(3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^ 2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c*arcsin(c*x) - (9*c^4*x^5 + 20 *c^2*x^3 + 120*x)/c^4)*b^2*d
Time = 0.34 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.71 \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=-\frac {1}{7} \, a^{2} c^{2} d x^{7} + \frac {1}{5} \, a^{2} d x^{5} - \frac {{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d x \arcsin \left (c x\right )^{2}}{7 \, c^{4}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} a b d x \arcsin \left (c x\right )}{7 \, c^{4}} - \frac {8 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d x \arcsin \left (c x\right )^{2}}{35 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d x}{343 \, c^{4}} - \frac {16 \, {\left (c^{2} x^{2} - 1\right )}^{2} a b d x \arcsin \left (c x\right )}{35 \, c^{4}} - \frac {{\left (c^{2} x^{2} - 1\right )} b^{2} d x \arcsin \left (c x\right )^{2}}{35 \, c^{4}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{49 \, c^{5}} + \frac {484 \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d x}{42875 \, c^{4}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )} a b d x \arcsin \left (c x\right )}{35 \, c^{4}} + \frac {2 \, b^{2} d x \arcsin \left (c x\right )^{2}}{35 \, c^{4}} - \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} a b d}{49 \, c^{5}} - \frac {16 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{175 \, c^{5}} - \frac {3358 \, {\left (c^{2} x^{2} - 1\right )} b^{2} d x}{385875 \, c^{4}} + \frac {4 \, a b d x \arcsin \left (c x\right )}{35 \, c^{4}} - \frac {16 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d}{175 \, c^{5}} + \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d \arcsin \left (c x\right )}{105 \, c^{5}} - \frac {37384 \, b^{2} d x}{385875 \, c^{4}} + \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d}{105 \, c^{5}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{35 \, c^{5}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} a b d}{35 \, c^{5}} \]
-1/7*a^2*c^2*d*x^7 + 1/5*a^2*d*x^5 - 1/7*(c^2*x^2 - 1)^3*b^2*d*x*arcsin(c* x)^2/c^4 - 2/7*(c^2*x^2 - 1)^3*a*b*d*x*arcsin(c*x)/c^4 - 8/35*(c^2*x^2 - 1 )^2*b^2*d*x*arcsin(c*x)^2/c^4 + 2/343*(c^2*x^2 - 1)^3*b^2*d*x/c^4 - 16/35* (c^2*x^2 - 1)^2*a*b*d*x*arcsin(c*x)/c^4 - 1/35*(c^2*x^2 - 1)*b^2*d*x*arcsi n(c*x)^2/c^4 - 2/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b^2*d*arcsin(c*x)/c ^5 + 484/42875*(c^2*x^2 - 1)^2*b^2*d*x/c^4 - 2/35*(c^2*x^2 - 1)*a*b*d*x*ar csin(c*x)/c^4 + 2/35*b^2*d*x*arcsin(c*x)^2/c^4 - 2/49*(c^2*x^2 - 1)^3*sqrt (-c^2*x^2 + 1)*a*b*d/c^5 - 16/175*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d *arcsin(c*x)/c^5 - 3358/385875*(c^2*x^2 - 1)*b^2*d*x/c^4 + 4/35*a*b*d*x*ar csin(c*x)/c^4 - 16/175*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*d/c^5 + 2/10 5*(-c^2*x^2 + 1)^(3/2)*b^2*d*arcsin(c*x)/c^5 - 37384/385875*b^2*d*x/c^4 + 2/105*(-c^2*x^2 + 1)^(3/2)*a*b*d/c^5 + 4/35*sqrt(-c^2*x^2 + 1)*b^2*d*arcsi n(c*x)/c^5 + 4/35*sqrt(-c^2*x^2 + 1)*a*b*d/c^5
Timed out. \[ \int x^4 \left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2 \, dx=\int x^4\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d-c^2\,d\,x^2\right ) \,d x \]